What is the value of exponential e? We know this value is varying between 2 and 3. But what is its actual value?
In mathematics, the constant e has a number of important properties. The brief answer to your question is that it has a value of
e=2.7182818284 (to ten decimal places)
It is defined as the function f(x)=e^x whose derivative at the point x=0 is exactly 1. That is, the slope of the tangent line at x=0 is 1. For other numbers, the slope of the tangent line is not exactly 1. Another definition of e is that it is the number given by the limit as n approaches infinity of
Its usefulness lies in its properties. For one, the derivative of
and for other exponential functions, the derivative of something like
Another useful property is its relationship to complex numbers. We define Euler's formula as
e^ix = cos x + i sin x
giving rise to trigonometric representations on a graph in the complex plane.
e , like pi, is the most interesting constant in mathematics.
the definition of e according to Mathematical Analysis is given by:
e = limit x-->infinity of the finction (1+1/x)^x, aternatively,
e = Limit x--> 0 (1+x)^(1/x), or
e = 1+1+ 1/2!+1/3!+1/4!+1/5!+....+1n!+........................, a positive valued positive termed decreasing series.
Its value is clearly greater than 2 as 1st 2 terms itself add up to 2. And the series becomes bounded above as:
e=1+1/1!+ 1/2!+1/3!+1/4!+1/5!+....+1n!+........................ is
< 1+1+1/2+1/2^2+1/2^3+1/2^4+1/2^5, as each terms in the former series is < each term in the latter series, except the 1st 3terms which are equal. But the latter adds up in limit to 1+(1-1/2)^(-1) = 1+2 = 3. So , 2 <e <3.
So e has definite value.
We can calculate approximate values of e from (1+1/n)^n or the series form at (iii) .
e is not a rational number. It is an irrational number.
e is not the root of any rational polynomial. It is not a surd.In other words, it is a transcendental number.
It is Leonard Euler, the Swiss mathematician who started calling the number by the name e. And now it is Euler,s number or Euler's constant.
e^(i*pi) = -1 Or e^(i*pi)+1 = 1 is one of the wonderous equations in Mathematics as it connect two famous transcental numbers e and pi and relates with the rational real number 1 or -1 and also imaginary number i or (-1)^(1/2). This equation is enlightened to us by Leonard Euler.
So the value of e is given by:
e = 1+1/1!+ 1/2!+1/3!+1/4!+1/5!+............ in infinite series form.
Also if you want in rational form the value of e is like:
e = 2. 7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274..